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Effective action and linear response of compact objects in Newtonian gravity

Sayan Chakrabarti, Térence Delsate, Jan Steinhoff

Abstract

We apply an effective field theory method for the gravitational interaction of compact stars, developed within the context of general relativity, to Newtonian gravity. In this effective theory a compact object is represented by a point particle possessing generic gravitational multipole moments. The time evolution of the multipoles depends on excitations due to external fields. This can formally be described by a response function of the multipoles to applied fields. The poles of this response correspond to the normal oscillation modes of the star. This gives rise to resonances between modes and tidal forces in binary systems. The connection to the standard formalism for tidal interactions and resonances in Newtonian gravity is worked out. Our approach can be applied to more complicated situations. In particular, a generalization to general relativity is possible.

Effective action and linear response of compact objects in Newtonian gravity

Abstract

We apply an effective field theory method for the gravitational interaction of compact stars, developed within the context of general relativity, to Newtonian gravity. In this effective theory a compact object is represented by a point particle possessing generic gravitational multipole moments. The time evolution of the multipoles depends on excitations due to external fields. This can formally be described by a response function of the multipoles to applied fields. The poles of this response correspond to the normal oscillation modes of the star. This gives rise to resonances between modes and tidal forces in binary systems. The connection to the standard formalism for tidal interactions and resonances in Newtonian gravity is worked out. Our approach can be applied to more complicated situations. In particular, a generalization to general relativity is possible.

Paper Structure

This paper contains 31 sections, 106 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Effective action
  3. Effective field theory in gravity
  4. Action for dynamical multipoles
  5. Quadrupole response
  6. Relativistic Love numbers
  7. Newtonian limit
  8. Potentials
  9. Action for stellar perturbations
  10. Variational principle for perturbations
  11. Normal modes
  12. Amplitude formulation
  13. Further developments
  14. STF basis
  15. Multipoles and overlap integrals
  16. ...and 16 more sections

Figures (1)

  • Figure 1: Quadrupolar $l=2$ response function for a star with $R = 8.89$ km, $m = 1.2 M_\odot$, and polytropic index 1 obtained numerically. The dots are just some selected data points. Many more were used for the fit, with higher density around the poles.